Isophote Fitting

When the task starts up you will be left in graphics cursor mode, at which point you can set the necessary first-guess values for the ellipse parameters. First, position the cursor in the image display at the desired center location for the first ellipse and then press . This will define parameters x0 and y0. You can then either position the cursor at the desired starting radius and position angle and press to redefine the a0 and teta parameters, or you can enter a value from the keyboard by typing :teta 0.0, for example. Finally, the initial value for eps is defined by entering a value from the keyboard, such as :eps 0.1. Before attempting the fit for the first ellipse you can review the current parameter settings by typing :show. If none of the displayed parameters have values of "INDEF" you are ready to proceed with the fitting.

To proceed with the fitting, you can press to compute a fit for the current isophote, press to proceed to the next isophote, or press to continue the fitting in non-interactive mode.

Using the initial ellipse values, the image is sampled along a first-guess elliptical path, producing a one-dimensional intensity distribution as a function of the ellipse eccentric angle. The harmonic content of this distribution is analyzed by fitting it to the following least squares function:

From the harmonic amplitudes (A1, B1, A2, B2), and from the local image gradient, correction factors can be derived for each sampling ellipse parameter (keeping a0 fixed). The parameter that corresponds to the largest amplitude is varied, and the image is resampled. This process is repeated until any one of the following criteria are met:

1. The largest harmonic amplitude is less than the internal variable crit times the rms residual of the intensity data around the harmonic fit. (The value of crit is defined below).

2. A user-specified maximum number of iterations, maxit, is reached.

3. More than 50% of the sampled points have the value INDEF. This usually happens because points on the elliptical sampling path that lie outside the image boundaries are given the value INDEF. However, badly flagged regions of an image can also cause the iteration loop to halt prematurely.

where n = 3 and n = 4. The amplitudes (A3, B3, A4, B4), when divided by the semi-major axis length and local intensity gradient, measure the isophote's deviations from perfect ellipticity--these parameters, all other associated isophote parameters, and some auxiliary information, are written to the output file.

After fitting the ellipse that corresponds to a given value of the semi-major axis (by the process described above), the axis value is increased by a factor of (1. + astep), and the process is repeated until either the semi-major axis value reaches the value maxa, or the last fitted ellipse has more than 50% of its sampled points flagged with INDEF values. The process then resumes from the first fitted ellipse (a0) inwards, in steps of (1./(1. + astep)), until the semi-major axis length reaches the value mina. At each step, the starting ellipse parameters are taken from the previously fitted ellipse that has the closest semi-major axis length.

On low surface-brightness regions (i.e., those having large radii), the small values of the image radial gradient can result in large corrections and meaningless values for the ellipse parameters. In this case, the iterative mode is disabled and geometric parameters keep their last stable values. This non-iterative mode is entered when one of the following criteria are met:

1. The absolute value of the image radial gradient is less than lslope times the rms along the isophote.

2. The ellipticity exceeds 1.0.

3. The ellipse center crosses the image boundaries.

4. The semi-major axis exceeds maxrit; maxrit can be ignored if its value is set to zero or INDEF.

If any sector vertex is outside the image boundary, its corresponding sample point is flagged as INDEF. The value of integrmode is ignored when the distance between the present ellipse and the next ellipse is less than 2 pixels, or when the semi-major axis is less than 20 pixels; in these cases, bi-linear interpolation is always used. The mean and median methods execute 3-5 times slower than bi-linear interpolation.

The basic convergency criterion (i.e., the crit variable), is defined as follows. In the case of bi-linear interpolation, it has a fixed value of 4%. In the case of either mean or median over elliptical annulus sectors, however, the increased number of pixels counted inside each sector as the algorithm proceeds outwards, will tend to decrease the rms residual against which harmonic amplitudes are compared. This will, on one side, increase the sensitivity to low-amplitude harmonics, and on the other side will increase the number of iterations necessary for the criterion to be met. The task parameter conver can be used to decrease the algorithm sensitivity in the outer regions of images, where sectors are large. The value of crit is defined as:

where S is the sector area (kept approximately constant along the ellipse) in units of pixel2. Thus conver=1.0 should, in principle, cancel the effect of the smaller rms residual that is caused by integration over larger areas, when compared with bi-linear interpolation. This will, however, retain all data for the full image area in the resulting sample. Larger values of conver will relax the criterion while smaller values will tighten it. If conver = 0.0, the crit parameter will be fixed at 4% regardless of the integration method used.

Errors for all parameters are obtained directly from the rms scatter of intensity data along the fitted ellipse.

For the sake of convenience, task output includes semi-major axis length raised to the 1/4 power, as well as isophotal intensities and errors expressed in a rough magnitude scale. This scale is derived from the parameters mag0, refer, and dark, and from the mean isophotal intensity int, according to the following relation: